An Optimization Method for Distribution Network Voltage Stability Based on Dynamic Partitioning and Coordinated Electric Vehicle Scheduling

Abstract

The integration of high-penetration renewable energy sources (RESs) and electric vehicles (EVs) increases the risk of voltage fluctuations in distribution networks. Traditional static partitioning strategies struggle to handle the intermittency of wind turbine (WT) and photovoltaic (PV) generation, as well as the spatiotemporal randomness of EV loads. Furthermore, existing scheduling methods typically optimize EV active power or reactive compensation independently, missing opportunities for synergistic regulation. The main novelty of this paper lies in proposing a spatiotemporally coupled voltage-stability optimization framework. This framework, based on an hourly updated electrical distance matrix that accounts for RES uncertainty and EV spatiotemporal transfer characteristics, enables hourly dynamic network partitioning. Simultaneously, coordinated active–reactive optimization control of EVs is achieved by regulating the power factor angle of three-phase six-pulse bidirectional chargers. The framework is embedded within a hierarchical model predictive control (MPC) architecture, where the upper layer performs hourly dynamic partition updates and the lower layer executes a five-minute rolling dispatch for EVs. Simulations conducted on a modified IEEE 33-bus system demonstrate that, compared to uncoordinated charging, the proposed method reduces total daily network losses by 4991.3 kW, corresponding to a decrease of 3.9%. Furthermore, it markedly shrinks the low-voltage area and generally raises node voltages throughout the day. The method effectively enhances voltage uniformity, reduces network losses, and improves renewable energy accommodation capability.

1. Introduction

1.1. Background

The advent of global warming and the energy crisis has compelled researchers worldwide to dedicate efforts to studying the application of various new energy sources [1,2]. In recent years, due to the environmentally friendly characteristics of electric vehicles (EVs), the number of EVs in use and the scale of supporting charging infrastructure have been rapidly growing [3], which has alleviated the energy shortage problem to some extent [4]. Meanwhile, EVs can also complement intermittent resources such as solar and wind power. Considering that EV charging loads exhibit randomness in both time and space, the traditional uncoordinated charging mode can impose various non-negligible negative impacts on the power grid, such as increased peak-valley differences, voltage drop, and feeder overload, potentially even threatening grid operational security [5,6]. Therefore, it is necessary to study the steady-state control of power grids with EV integration from the perspective of power control, including both active and reactive power control. With the continuous advancement of charging infrastructure, vehicle-to-grid (V2G) technology has gradually become one of the important forms of vehicle–grid interaction [7]. Specifically, considering the adjustability of user charging behavior and the spatio-temporal transfer characteristics of EV loads, EVs can participate in regional grid reactive power optimization as a reactive power source, providing the grid with diverse ancillary services such as peak shaving and voltage regulation, thereby mitigating the negative impacts on the grid [8,9,10].

1.2. Literature Review

The large-scale integration of EV loads into distribution networks introduces high spatio-temporal uncertainty, which can exacerbate localized voltage fluctuations and increase the risk of voltage violations. To address this issue, several studies have been conducted from multiple perspectives [11]; however, conventional regulation methods still exhibit certain limitations in terms of response speed and spatial adaptability.

Traditional distribution network voltage control primarily relies on reactive power regulation [12], such as adjusting the tap positions of on-load tap changers (OLTC) or switching capacitor banks in and out [13]. However, the response speed of the above methods is relatively slow, with high operation and maintenance costs, which is not suitable for frequent actions. Simultaneously, the reactive power compensation devices configured for traditional regulation are limited in numbers and fixed locations, rendering them ineffective in counteracting voltage fluctuations induced by EVs across spatial scales. Considering the increasing complexity of power grids, some scholars have proposed that partitioning the distribution network can help narrow down the search scope for locating reactive power compensation devices, thereby achieving precise reactive power and voltage regulation. For example, Ref. [14] proposed a sensitivity analysis method, which is, when a node shows a tendency for a voltage anomaly, it first identifies several associated nodes that most significantly affect that node’s voltage. Then, customized economic incentives are executed based on the actual contribution of EVs at these associated nodes to voltage regulation until the voltage issue at the target node is resolved. However, this method focuses on enhancing the willingness of EVs to participate in voltage control—namely, setting customized economic incentives to make EV owners at specific nodes more willing to participate in voltage control. However, these approaches lack consideration of the spatio-temporal migration characteristics of EVs.

Due to the evident spatio-temporal regional distribution characteristics of EVs [15], distribution network partitioning becomes one of the effective methods to describe such characteristics. Ideally, after partitioning, the nodes within each sub-region of the distribution network exhibit strong coupling, while nodes outside a sub-region are weakly coupled with the nodes inside it. Electrical distance [16] and modularity index [17] are standard metrics commonly used for network partitioning. Ref. [18] proposed a partitioning method based on weights reflecting sensitivity to reactive power and active power losses, aiming to maximize the utilization of reactive power compensation devices. Ref. [19] proposed a network partitioning method considering PV uncertainty and incorporating the influence of active power injections at nodes on voltage magnitude into the metrics used for partitioning. This method leads to a more reasonable network division, but still has certain limitations, such as insufficient consideration of generation resource types and an incomplete formulation of the electrical distance index. Ref. [20] considered factors such as power balance during the partitioning process. Ref. [21] developed a partitioning model based on an index system comprising correlation coefficients between nodes, cluster coupling coefficients, and comprehensive coupling coefficients. With the optimization objective of maximizing the comprehensive index, it aimed to find a specific distribution network node partition for each PV unit, i.e., a partition where node voltages are most responsive to reactive power variations from that specific PV unit and less responsive to variations from other PV units. Although the aforementioned literature proposes various distribution network partitioning methods, most existing studies primarily rely on static electrical distance or fixed RES output scenarios. They fail to adequately account for the real-time impact of wind and solar intermittency as well as the spatiotemporal transfer characteristics of EVs under high-RES-penetration or high-EV-penetration scenarios. Consequently, current partitioning strategies struggle to adapt to the time-varying characteristics of grid loads. How to effectively partition distribution networks with large-scale EV integration remains a critical research issue that requires further investigation.

In addition to the aforementioned shunt capacitor banks and OLTCs [22,23], scholars have also investigated the application of coordinated voltage and var control (VVC) in distribution networks, involving intelligent devices (e.g., smart inverters) working alongside traditional reactive power regulation devices [24]. The participation of EVs in grid ancillary services is primarily realized through vehicle-to-grid (V2G) technology. Through V2G technology, EVs can actively provide both active and reactive power to the grid, thereby offering ancillary services and further leveraging the bidirectional regulation advantage of EVs [25]. As a resource capable of bidirectional charging and discharging, extensive studies have been conducted on the active power regulation of EVs [26,27,28]; however, their potential in reactive power regulation and distribution network voltage support remains unexplored.

The reactive power regulation capability of EV chargers has not yet been widely utilized. In contrast, EV charging has long been regarded as a source of reactive power disturbance, requiring operation under limited power factor conditions, which largely constitutes a waste of resources. The electronic power converters, as a component of EV charging equipment, can act almost instantaneously due to their inherent characteristics, protecting sensitive loads and devices from grid voltage variations. Simultaneously, the hardware of EV chargers can inject reactive power into the grid to support voltage regulation [29,30] without degrading battery lifespan [31]. EV charging stations also possess the capability to absorb or inject reactive power. Ref. [32] used a differential evolution algorithm to optimize the reactive power dispatch of EV charging stations, combined with a two-point estimation method to handle uncertainties from renewable energy, loads, and EV demand. However, uncoordinated injection of reactive power into the grid can instead be detrimental to grid voltage [33]. Therefore, Ref. [34] proposed a hybrid VVC framework based on a hierarchical control method for highly unbalanced distribution networks. This method can aggregate reactive power injected by geographically proximate EVs while reducing communication delays, effectively decreasing the response time. A similar hierarchical control framework was also applied in [35].

Due to the high resistance-to-reactance (R/X) ratio of distribution lines, voltage regulation relying solely on reactive power adjustment struggles to address significant voltage fluctuations in distribution networks. In this case, achieving effective voltage control requires a coordinated consideration of both active and reactive power distribution within the distribution network [36]. Ref. [36] proposed an integrated reactive power optimization method that fully accounted for the travel uncertainty and charging demands of EVs, achieving coordinated distributed EV charging and rapid V2G-based reactive power response. Ref. [37] proposed a centralized voltage control scheme that optimized the active and reactive power of EVs and other controllable loads within the region. Ref. [38] coordinated multiple resources for distribution network voltage regulation. Based on forecasts of PV output, load, and EV driving patterns, it made decisions to utilize the active and reactive power regulation capabilities of EVs to improve voltage profiles and enhance energy-saving potential. These studies provide feasible approaches for the joint optimization of active and reactive power in distribution networks. However, they often rely on day-ahead forecasting or a centralized architecture, hindering real-time response to system fluctuations and uncertainties.

With its core mechanisms of predictive control, rolling optimization, and feedback correction, model predictive control (MPC) provides a feasible framework to address the control challenges posed by large-scale RES integration [39]. The rolling-horizon optimization characteristic of MPC enables continuous dynamic adjustment of control decisions, thereby allowing more precise coordination of active and reactive power from distributed resources like EVs to achieve effective and rapid voltage control. To leverage large-scale distributed EVs, Ref. [40] proposed a distributed MPC framework applicable to both balanced and unbalanced distribution networks. However, this framework did not consider the active power regulation of EVs. Ref. [41] introduced a credibility index to assess the regularity of user driving behavior, subsequently formulating corresponding charging and pricing strategies to reduce day-ahead and intra-day electricity consumption errors. But this method did not consider the reactive power regulation capability of EVs. Ref. [42], based on the MPC method, proposed a computationally efficient and privacy-preserving decentralized coordination framework between distribution system operators and charging station operators. It aimed to achieve robust operation of the distribution network and EV charging stations under uncertainties in active three-phase unbalanced distribution systems. However, this method did not fully explore the reactive power regulation potential of EVs, and each charging station operator only managed the EVs within its own station, failing to utilize the spatial mobility of EVs. The aforementioned methods have made valuable explorations in addressing the uncertainties brought by EV integration. However, most methods fail to synergistically utilize the bidirectional active and reactive power regulation capabilities of EVs. The methods employed in the aforementioned studies have made valuable explorations in addressing the uncertainties introduced by EV integration. However, most existing research primarily focuses on the active power scheduling of EVs. Regarding reactive power optimization, studies largely concentrate on the reactive compensation capability of EV chargers, yet insufficient attention has been paid to the co-optimization of active power dispatch and reactive compensation. This limitation restricts the potential of EVs to serve as flexible reactive power sources for enhancing voltage stability. Furthermore, existing research often treats the reactive power compensation capability of EVs as an add-on feature independent of active power scheduling. In reality, the available capacity for reactive power in EVs is constrained by the state of charge (SOC) and the active power draw. Focusing solely on reactive power regulation may lead to an overestimation of its reactive potential, which could subsequently impact grid operation.

1.3. Paper Contributions and Organizations

The main objective of this research is to develop a closed-loop voltage-stability optimization method for active distribution networks with high RES and EV penetration that (i) adapts network partitioning to time-varying operating conditions and (ii) coordinates EV active and reactive power in real time to improve voltage profiles and reduce network losses under EV/RES uncertainty and operational constraints.

To achieve this objective, the main contributions are as follows:

• A dynamic partitioning strategy based on hourly updates to the electrical-distance matrix is proposed, incorporated with PV output, WT output, and EV dispatchable capacity. The hourly adaptive partitioning model is capable of enhancing the grid’s adaptability to RES fluctuations and EV spatiotemporal transfer characteristics.
• Leveraging the bidirectional charging/discharging capability of EVs, reactive power response is coordinated with active power scheduling. By regulating the power-factor angle of the bidirectional charger, joint active–reactive control is achieved, improving voltage support while respecting SOC-dependent capability limits.
• A hierarchical optimization framework based on model predictive control (MPC) is designed. The upper layer updates partitions and identifies critical voltage nodes on an hourly basis using forecasting WT/PV output and EV availability. The lower layer performs 5 min rolling optimization of EV charging/discharging and reactive support setpoints to enable real-time power allocation and spatiotemporal coordination.

The structure of this paper is organized as follows: Section 2 introduces the overall framework and problem statement. Section 3 presents the dynamic partitioning method. Section 4 focuses on constructing the optimization model and the hierarchical MPC framework. Section 5 demonstrates the optimization results, and finally, Section 6 presents the conclusions.

2. General Description and Problem Statement

Figure 1 illustrates the proposed closed-loop voltage stability optimization framework for active distribution networks with high-penetration renewable energy and electric vehicle integration. This framework integrates dynamic network partitioning with coordinated EV active-reactive power scheduling and employs a hierarchical model predictive control architecture, achieving synergy between dynamic partitioning in the spatial dimension and rolling scheduling in the temporal dimension. The dynamic partitioning aims to capture the time-varying coupling relationships between nodes induced by PV/wind turbine output fluctuations and EV uncertainties. The coordinated scheduling leverages the bidirectional charging/discharging capability and reactive power support potential of EVs to provide the system with rapid voltage regulation. The framework is based on multi-source information inputs, including distribution network topology and parameters, real-time measurement data (node voltages, branch power flows, and loads), short-term renewable energy output forecasts, EV arrival/departure times and their state-of-charge (SOC) information, as well as time-of-use electricity prices. Based on this, the framework outputs adaptive partitioning schemes and key voltage nodes within each partition, and generates detailed control commands for each EV charging point, including setpoints for active charging/discharging power and reactive power.

The framework comprises four tightly coupled functional modules. The multi-source data acquisition and forecasting module is responsible for integrating real-time measurements and short-term forecast information, providing data support for the upper-layer partitioning and scheduling decisions. The upper-layer dynamic partitioning and key node identification module operates on an hourly basis. Relying on the real-time updated electrical distance matrix, it comprehensively considers the spatiotemporal distribution of renewable energy output fluctuations and EV dispatchable capacity to calculate the coupling strength between nodes. Combined with a partition capacity imbalance index, it dynamically divides the network into regions and subsequently identifies the node with the maximum voltage deviation within each partition as the key node, thereby providing spatial-dimensional guidance for the lower-layer scheduling.

The lower-layer rolling EV active/reactive power coordinated scheduling module performs rolling optimization with a 5 min cycle. This module fully exploits the potential of EVs as flexible bidirectional resources. By adjusting the power factor angle of the three-phase six-pulse charger and subject to constraints such as battery SOC dynamics, charging/discharging power limits, and user travel demands, it coordinates and optimizes the active and reactive power output of EVs. The optimization objectives balance minimizing voltage deviation and reducing network losses from the grid side with minimizing charging costs from the user side, aiming to reconcile system secure operation and user benefits. The real-time execution and feedback correction module dispatches the optimized commands to the charging points, verifies the control effectiveness through power flow calculations, and feeds the updated system state back to the next cycle, forming a closed-loop correction mechanism.

Through the hierarchical MPC structure, the hourly partitioning decisions from the upper layer provide stable regional boundaries and regulation focuses for the minute-level power scheduling in the lower layer, enabling the spatial decomposition and regulation of rapidly fluctuating renewable energy and EV loads. Conversely, the lower-layer rolling optimization can track system state changes in real-time, and its scheduling results further provide the basis for updating the electrical distance matrix and evaluating partitioning effectiveness, forming a spatiotemporally coupled coordination mechanism. The proposed method is capable of addressing the long-term coupling relationship changes introduced by high-penetration renewable energy integration and the uncertainties arising from the stochastic behavior of EVs. It offers a solution for active distribution network voltage stability control that possesses both spatial adaptability and temporal responsiveness. Subsequent chapters will elaborate in detail on the mathematical models and solution processes of each module, including the dynamic partitioning method, the coordinated EV active/reactive power optimization model, and the implementation process of the hierarchical model predictive control.

3. Dynamic Partitioning Method for Distribution Networks Considering WT/PV Fluctuations and EV Dispatchable Capacity

To address the challenges to distribution network voltage stability posed by high-penetration RES and EVs, this chapter proposes a dynamic partitioning method that considers wind/PV fluctuations and EV dispatchable capacity. Specifically, it captures the time-varying characteristics of system operational states through a real-time updated electrical distance matrix. Based on this, the method comprehensively considers the spatio-temporal transfer characteristics of EVs to ultimately achieve hourly adaptive adjustment of distribution network partitions. This section first describes the electrical distance metric used for partitioning. Then, it analyzes RES output uncertainty. Finally, it integrates these factors to elaborate the dynamic partitioning adjustment mechanism that accounts for the dispatchable spatiotemporal distribution of EVs.

3.1. Electrical Distance Definition

Electrical distance can clearly describe the relationship between power injection and voltage variation, as well as characterize the coupling relationship and degree of mutual influence between nodes. It is highly related to the topology of the distribution network. In distribution network partitioning, the electrical distance can be derived by inverting the Jacobian matrix of the distribution network. Accurate electrical distance significantly impacts the quality of partition delineation. Here, the Jacobian matrix J describes the linear relationship between node power injections and voltage variations, which can be expressed as Equation (1):

$$\left[{\displaystyle \frac{\Delta P}{\Delta Q}}\right]=J\left[{\displaystyle \frac{\Delta \Theta }{\Delta V}}\right]$$

(1)

where $\Delta P$ and $\Delta Q$ represent the variations in active power and reactive power injections at the node, respectively; $\Delta \Theta$ and $\Delta V$ denote the variations in voltage phase angle and voltage magnitude at the node, respectively. Therefore, matrix S, which represents the impact of power variations on voltage variations, can be obtained by inverting the Jacobian matrix, as Equation (2):

$$S=J^{-1}=\left[\begin{array}{cc}{S}_{\Theta ,P}& S_{\Theta ,Q}\\ S_{V,P}& S_{V,P}\end{array}\right]$$

(2)

accordingly, the voltage variation vector can be written explicitly in terms of the power-injection variation vector as Equation (3):

$$\left[{\displaystyle \frac{\Delta \Theta }{\Delta V}}\right]=\left[\begin{array}{cc}{S}_{\Theta ,P}& S_{\Theta ,Q}\\ S_{V,P}& S_{V,Q}\end{array}\right]\left[{\displaystyle \frac{\Delta P}{\Delta Q}}\right]$$

(3)

here, $S_{\Theta ,P}$ and $S_{\Theta ,Q}$ denote the sensitivities of voltage phase-angle variations to active power and reactive power injection variations, respectively; $S_{V,P}$ and $S_{V,Q}$ denote the sensitivities of voltage-magnitude variations to active- and reactive-power injection variations, respectively. Based on this, the electrical distances $d_{ij}^P$ and $d_{ij}^Q$ between node i and node j can be expressed as Equations (4) and (5):

$$d_{ij}^P=v_p^{ii}+v_p^{jj}-v_p^{ij}-v_p^{ji}$$

(4)

$$d_{ij}^Q=v_q^{ii}+v_q^{jj}-v_q^{ij}-v_q^{ji}$$

(5)

here, $v_p^{ij}$ represents the sensitivity of voltage change at node i to active power injection at node j, and $v_q^{ij}$ represents the sensitivity of voltage change at node i to reactive power injection at node j. They are specifically expressed as Equations (6) and (7):

$$v_p^{ij}={\displaystyle \frac{\mathit{\partial }{U}_i}{\mathit{\partial }{P}_j}}$$

(6)

$$v_q^{ij}={\displaystyle \frac{\mathit{\partial }{U}_i}{\mathit{\partial }{Q}_j}}$$

(7)

since the sensitivities of voltage to active power and to reactive power between nodes depend on factors such as the network topology and load types, there is no simple proportional relationship between them, and they are relatively independent. Therefore, the inter-node electrical distance $d_{ij}$ is defined as a weighted value of the electrical distance $d_{ij}^P$ based on active power sensitivity and the electrical distance $d_{ij}^Q$ based on reactive power sensitivity. Can be expressed as Equation (8):

$$d_{ij}=\tau d_{ij}^P+(1-\tau )d_{ij}^Q$$

(8)

here, $\tau$ is the weight coefficient, representing the ratio of the impact of active power regulation to that of reactive power compensation on voltage when a voltage violation occurs. Its value can be adjusted and calculated based on different scenarios.

After normalization, the electrical distance partition index is defined as Equation (9):

$$L_{ij}=\sqrt{{\displaystyle \sum _{k=1}^n{({d^{\prime }}_{ik}-{d^{\prime }}_{jk})}^2}}$$

(9)

where $L_{ij}$ calculates the degree of difference between the responses of node i and node j to voltage changes at other nodes. A smaller $L_{ij}$ indicates that node i and node j have similar responses when influenced by voltage changes at other nodes.

3.2. Multi-Source Dynamic Data Input

The dynamic partitioning mechanism relies on accurate monitoring of the real-time status of the distribution network and forecasting controllable resources. It requires the fusion of dynamic data streams, including the real-time varying electrical distance matrix influenced by WT/PV output fluctuations, and the dispatchable capacity of EVs characterized by spatio-temporal distribution.

3.2.1. Renewable Energy Output Model

Compared to conventional thermal power, PV and WT outputs exhibit significant randomness and intermittency. As their penetration rate increases, power flows are significantly impacted, potentially causing frequent reversals and magnitude changes in feeder power flow directions. These variations directly affect the Jacobian matrix, causing the electrical distance, which characterizes node coupling relationships, to exhibit time-varying characteristics. Traditional fixed partitioning methods, based on static network design, struggle to adapt to the hourly fluctuations of the electrical distance matrix. Thus, it is essential to capture the impact of WT/PV output fluctuations on the network topology by updating the electrical distance matrix in real-time.

This section derives the output of photovoltaic power stations and wind farms based on typical daily solar irradiance and wind speed.

(1)

PV output Model

PV output generally depends on solar irradiance intensity and temperature conditions. However, the influence of temperature on photovoltaic power stations is usually limited; therefore, only the impact of solar irradiance is considered. The specific relationship is as in Equation (10):

$$P_{PV}=L{S}_{PV}\lambda$$

(10)

among them, $P_{PV}$ represents the photovoltaic output power, $L$ denotes the solar irradiance, $S_{PV}$ is the effective area of the photovoltaic panels, and $\lambda$ stands for the total system efficiency. The uncertainty of photovoltaic output power can be described by a Beta distribution, expressed as Equation (11):

$$f(P_{PV})={\displaystyle \frac{\Gamma (\alpha +\beta )}{\Gamma (\alpha )\Gamma (\beta )}}{({\displaystyle \frac{P_{PV}}{P_{PV,\max }}})}^{\alpha -1}{(1-{\displaystyle \frac{P_{PV}}{P_{PV,\max }}})}^{\beta -1}$$

(11)

where $P_{pv,\max }$ is the maximum possible output power of the photovoltaic system; $\alpha$ and $\beta$ are the shape parameter and the scale parameter, respectively.

(2)

WT output Model

The active power output of a wind farm is determined by wind speed, and its short-term wind speed variations follow a Gaussian distribution. Wind speed variations follow a normal distribution as Equation (12):

$$f(v)={\displaystyle \frac{1}{\sigma \sqrt{2\pi }}}\exp [-{\displaystyle \frac{{(v-\mu )}^2}{2{\sigma }^2}}]$$

(12)

where $v$ is the wind speed, $\mu$ and $\sigma$ are the mean and standard deviation of wind speed, respectively. Its active power output can be expressed as Equation (13):

$$P_W=\left\{\begin{array}{cc}0& V_w\le V_{ci}\\ P_{wr}{\displaystyle \frac{{\left(V_w-V_{ci}\right)}^3}{{\left(V_r-V_{ci}\right)}^3}}& V_{ci}\le V_w\le V_r\\ P_{wr}& V_r\le V_w\le V_{co}\\ 0& V_w>V_{co}\end{array}\right.$$

(13)

where $V_{ci}$ is the cut-in wind speed of the wind turbine; $V_r$ is the rated wind speed of the wind turbine; $V_{co}$ is the cut-out wind speed of the wind turbine; and $P_{wr}$ is the rated power of the wind turbine.

3.2.2. Electric Vehicle Stochastic Model

As EVs exhibit spatio-temporal transfer characteristics, simulating EV charging behavior requires considering the impacts of user travel patterns and battery characteristics. The household travel survey is a useful tool for analyzing the stochastic behavior of EVs [42]. Based on the U.S. 2017 national household travel survey, and assuming each EV connects to the grid once daily, the arrival time, departure time, arrival SOC, and departure SOC of EVs at charging nodes are generated from a probability distribution. The arrival time follows a mixed normal distribution, as shown in Equation (14):

$$f_{arr}(t)=\left\{\begin{array}{cc}{\omega }_1^{EV}{\displaystyle \frac{1}{{\sigma }_s\sqrt{2\pi }}}\exp [-{\displaystyle \frac{{(t-{\mu }_s)}^2}{2{\sigma }_s^2}}]& t\in [{\tau }_{EV},24)\\ {\omega }_2^{EV}{\displaystyle \frac{1}{{\sigma }_s\sqrt{2\pi }}}\exp [-{\displaystyle \frac{{(t+24-{\mu }_s)}^2}{2{\sigma }_s^2}}]& t\in [0,{\tau }_{EV})\end{array}\right.$$

(14)

since the arrival times of EV users typically exhibit a bimodal characteristic, the mixed normal distribution can better capture this pattern. Here, ${\mu }_s$ represents the central moment of user arrival time, with a value of 18.3; ${\sigma }_s$ describes the dispersion of user arrival time, with a value of 4.1; ${\tau }_{EV}$ is the time split point, dividing the day into two intervals to endow the mixed normal distribution with a bimodal characteristic, with a value of 6.3; ${\omega }_1^{EV}$ and ${\omega }_2^{EV}$ are normalization constants ensuring their sum equals 1; $t\in [0,24)$.

The departure time follows a normal distribution, as shown in Equation (15):

$$f_{dep}(t)={\displaystyle \frac{1}{{\sigma }_d\sqrt{2\pi }}}\exp [-{\displaystyle \frac{{(t-{\mu }_d)}^2}{2{{\sigma }_d}^2}}]$$

(15)

EV user departure times are generally concentrated in the morning; therefore, a unimodal normal distribution is used to capture this pattern. Here, ${\mu }_d$ represents the central moment (mean) of user departure time, with a value of 7.83; ${\sigma }_d$ describes the dispersion of user departure time, with a value of 1.5.

The arrival SOC and departure SOC of EVs follow a piecewise uniform distribution over K SOC intervals, as shown in Equation (16):

$$\begin{array}{cc}{f}_{SOC}(s)={\displaystyle \frac{p_k}{100\Delta s}}& s\in \left[s_{k-1},s_k\right],k=1,2,\cdots K\end{array}$$

(16)

where $K$ is the number of intervals, $\Delta s$ is the interval width, $p_k$ is the probability weight of the k-th interval, and ${\displaystyle {\sum }_{k=1}^K{p}_k}=1$. The arrival SOC of EV users shows a concentration at low SOC levels, while the departure SOC is concentrated at high SOC levels.

3.3. Dynamic Partitioning Adjustment Mechanism for Distribution Networks

To ensure each partition has sufficient adjustable resources (i.e., EVs) and to promote the adaptability of the partitioning scheme to the spatio-temporal transfer characteristics of EVs, this section incorporates the dispatchable capacity of EVs as one of the factors for dynamic partition adjustment. The distribution network is initially divided into three regions: residential, industrial, and commercial. The number of EVs in each region varies significantly across time periods, with notable differences existing between different regions during the same time period. Therefore, this dynamic adjustment can be expressed as Equation (17):

$$\min \psi (t)={\displaystyle \frac{1}{L_{total}}}{\displaystyle \sum _{i=1}^k\left|L_i(t)-{\beta }_i(t)\cdot \left|P_i\right|\right|}$$

(17)

where $\psi (t)$ is the partition capacity imbalance index; $P_i$ represents the partition result, and $\left|P_i\right|$ denotes the number of nodes within the partition; $L_i(t)$ is the actual dispatchable capacity of partition $P_i$ at time t, ${\beta }_i(t)\cdot \left|P_i\right|$ denotes the theoretical dispatchable benchmark value for partition $P_i$ at time t, and ${\beta }_i(t)$ is the EV time concentration factor, which is based on expert experience and is related to the EV distribution in each region across different time periods. Therefore, $\psi (t)$ can be intuitively interpreted as the magnitude of the difference between the actual dispatchable capacity and the theoretical benchmark value for each partition. The specific implementation of dynamic partitioning is shown in Algorithm 1. This algorithm is executed hourly, adjusting node affiliations to output an optimal partition scheme adapted to the system state for the upcoming hour.

Algorithm 1. Dynamic Partitioning of Distribution Network

Input:  $P_{PV}$, $P_{WT}$, EV distribution, ${\beta }_i(t)$ Output: Optimized partitioning scheme $P_{i\_optimized}$ 1:    for each time window t (hourly): 2.:   Calculate power flow based on latest RES and load data to update $L_{ij}$ 3:    initialize partitions $P_i$ 4:    while i < max_i + 1 and not converged: 5:      Compute $L_i(t)$ for all partitions 6:      Calculate $\psi (t)$ 7:      for each overloaded $P_i$: 8:        Locate centroid $C_i$ 9:        Identify boundary nodes $B_i$ 10:      Select $v^{\ast }$ = argmax $M(v)$ 11:      Find target $P_i^{\ast }$ minimizing $d(v^{\ast },C_i)$ 12:      if constraints satisfied: 13:       Migrate $v^{\ast }$ from $P_i$ to $P_i^{\ast }$ 14:      end if 15:    end for 16:          Update ${\beta }_i(t+1)$ 17:          Check convergence 18:     end while 19:  return optimized partitions $P_{i\_optimized}$ 20:  end for

In the table, $M(v)$ denotes the maximum normalized distance from node $v$ to the cluster center, specifically expressed as Equation (18):

$$M(v)={\displaystyle \frac{d(v,C_i)}{d_{\max }^i}}$$

(18)

it should be noted that $C_i$ represents the cluster center of a single partition; $B_i$ denotes the boundary nodes of a single partition. Furthermore, node adjustment must satisfy the constraint that each partition after migration remains a connected subgraph.

4. Optimization Scheduling Strategy

The uncertainty in distributed RES generation may lead to voltage fluctuations in the distribution network. Treating EVs as flexible resources for distribution network scheduling can not only effectively mitigate the aforementioned issues but also allow EV owners to receive corresponding economic compensation for their participation in regulation [43]. Thus, based on the dynamic partitioning strategy for distribution networks described in the previous section, coordinating EV resources within each region enables rapid, optimal control of the overall voltage, thereby enhancing the stability and reliability of the distribution network.

4.1. EV Charging/Discharging and Reactive Power Response Characteristics

4.1.1. Reactive Power Response of Charging Piles

It has been demonstrated that providing reactive power during EV charging does not affect the battery charging process or lifespan [44,45]. This characteristic allows EVs to act as controllable resources for reactive power compensation during charging, thereby supporting distribution network voltage.

The core of the V2G charger is a three-phase two-level bidirectional AC/DC converter (six-switch bridge), as shown in Figure 2. The grid is represented by three phase sources $v_{G1}–v_{G3}$. Each phase is connected to the converter through a filter inductor $L_1–L_3$ and an equivalent series resistance $R_1–R_3$, which limit current ripple and shape the grid currents. The converter consists of three bridge legs: the upper fully controlled switches $T_1–T_3$ and the lower fully controlled switches $T_4–T_6$. Each switch has an anti-parallel diode $D_1–D_6$ to provide a reverse-current path. A DC-link capacitor $C$ stabilizes the DC-bus voltage and is connected to the EV battery/load.

The filter inductors $L_1–L_3$ store energy and suppress high-frequency switching ripple so that the grid currents can be regulated to near-sinusoidal waveforms; $R_1–R_3$ model the equivalent conduction losses of the filter and line. The six controlled switches $T_1–T_6$ are modulated by PWM to synthesize the converter phase voltages $v_{O1}–v_{O3}$ and to track commanded phase currents, thereby controlling active power and reactive power. In each bridge leg, the upper device $T_1–T_3$ connects the corresponding phase node to $+v_{dc}$, and the lower device $T_4–T_6$ connects it to $-v_{dc}$; upper and lower switches in the same leg are driven in a complementary manner with dead-time to avoid DC-bus shoot-through. The anti-parallel diodes $D_1–D_6$ provide the necessary freewheeling/commutation paths for the inductive currents, enabling bidirectional current flow. The DC-link capacitor $C$ buffers the instantaneous power mismatch between AC and DC sides and reduces DC-bus voltage ripple. In G2V charging, the converter operates as an active rectifier; in V2G discharging, it operates as an inverter while still being able to provide reactive power support [44].

Under steady-state operation, the voltage drops across the line impedance, $v_{RL}$, is determined by the vector difference between the grid voltage $v_G$ and the converter voltage $v_O$. Consequently, this can alter both the magnitude and phase of the supply current $i$ flowing from the grid to the charger. Since the grid voltage $v_G$ is approximately constant, the system’s complex power can be controlled by changing the current $i$. Therefore, by adjusting $v_O$ to modify $v_{RL}$ and $i$, bidirectional flow of both active power and reactive power can be achieved. The specific vector diagram is shown in Figure 3.

Since the converter must operate within its rated current limit and $v_{RL}$ is also constrained by its rated value, the controllable range of the converter voltage is represented by the dashed circle in Figure 3. Simultaneously, the active power and reactive power transmitted by the bidirectional converter are constrained by the maximum apparent power, as shown in Equation (19):

$$S_{\max }^2\ge P^2+Q^2$$

(19)

where $S_{\max }$ is the maximum apparent power of the converter; $P$ and $Q$ are the active power and reactive power flowing through the converter, respectively. The specific relationship is illustrated in Figure 4:

The reactive power regulation capability of an EV is constrained by the maximum apparent power and the magnitude of active power during charging or discharging. As shown in Figure 4, if the charging power is $P_o$, the corresponding reactive power response range $Q_{range}$ is illustrated in the figure.

This is adjusted via the power factor angle $\phi$. The converter’s active and reactive power can be parameterized by the apparent power $S$ and $\phi$ as Equation (20):

$$\left\{\begin{array}{c}P=S\cos \phi \\ Q=S\sin \phi \end{array}\right.$$

(20)

the power factor angle is constrained by Equation (21):

$${\phi }_{\min }\le \phi \le {\phi }_{\max }$$

(21)

4.1.2. Individual Electric Vehicle Charging-Discharging Model

EVs can be integrated into the distribution network as distributed energy storage units. Within the constraints of charging and discharging power constraints, bidirectional power exchange with the grid can be achieved through V2G technology. During charging, it acts as an load for electricity consumption; during discharging, it serves as a mobile power source, supplying electricity to the grid. Specifically, when participating in grid scheduling, the charging/discharging power and capacity model of an EV is illustrated in Figure 5.

In the figure, point a represents the EV connecting to the grid at time $t_s$ with an initial SOC of $SO{C}_s$. The slope of line segment ab corresponds to the maximum charging power of the EV, and the slope of line segment a-g corresponds to the maximum discharging power. The boundary line a-c-d indicates that the EV starts charging at the maximum power immediately after connecting to the grid and stops when the state of charge reaches the maximum level. The boundary line a-g-f-e indicates that the EV starts discharging at the maximum power immediately after connecting and stops when the SOC reaches $SO{C}_{\min }$. Since the model enforces the constraint that the EV must reach at least the desired state of charge $SO{C}_e$ by the departure time $t_e$, the system triggers mandatory charging when it detects that following the current discharging or idle schedule would fail to achieve the desired SOC before disconnection. The dashed line directly connecting points a and e in the figure represents a possible charging path.

4.2. Optimization Model

4.2.1. Objective Function

The proposed regulation strategy is based on the following three objectives: minimizing voltage deviation, grid network losses, and the charging cost for EV owners. The specific calculation methods for these objectives are as follows:

(1)

Voltage Deviation

$$\min f_1={\displaystyle \sum _{t=1}^T{\displaystyle \sum _{i=1}^{N_{bus}}\left|V_{i,t}-V_{ref}\right|}}$$

(22)

where $f_1$ represents the total system voltage deviation, $V_{i,t}$ is the voltage magnitude at node i at time t; $T$ is the total number of optimization periods; $N_{bus}$ is the number of network nodes; and $V_{ref}$ is the ideal voltage reference value.

(2)

Network Losses

$$\min f_2={\displaystyle \sum _{t=1}^T{\displaystyle \sum _{l=1}^{l_{\max }}\left(r_l{I}_l{(t)}^2\right)}}$$

(23)

where $f_2$ represents the distribution network losses; $l_{\max }$ is the total number of network branches; $r_l$ is the resistance of branch $l$; and $I_l(t)$ is the current flowing through branch $l$ during period t.

(3)

EV Owner Charging Cost

The charging cost affects the willingness of EV owners to participate in scheduling. When the charging cost is significantly reduced or the discharging revenue is sufficiently high, EV owners are more inclined to participate in scheduling. It is specifically expressed as Equation (24):

$$\min f_3={\displaystyle \sum _{t=1}^T{\displaystyle \sum _{k=1}^{E{V}_{\max }}\left({\lambda }_{c,t}{P}_{c,t,k}-{\lambda }_{disc,t}{P}_{disc,t,k}+C_{\mathrm{deg}}-C_{Q,k}\right)}}$$

(24)

where $f_2$ represents the total charging cost for EV owners; $E{V}_{\max }$ is the total number of EVs; ${\lambda }_{c,t}$ and ${\lambda }_{disc,t}$ are the time-of-use charging and discharging electricity prices for period t, respectively; $P_{c,t,k}$ and $P_{disc,t,k}$ are the active charging and discharging power of the k-th EV during period t, respectively; and $C_{\mathrm{deg}}$ is the battery degradation cost, which is specifically expressed as Equation (25):

$$C_{\mathrm{deg}}=\gamma {\eta }_{\mathrm{deg}}\left|P_{disc,t,k}\right|$$

(25)

where $\gamma$ is the battery replacement cost; ${\eta }_{\mathrm{deg}}$ is the battery degradation coefficient per unit of discharged energy. $C_{Q,k}$ represents the reactive power response subsidy for EVs, whose benefit from reduced network losses is reflected in the decrease in grid active power demand. The unit reactive-power compensation price is calculated as shown in Equation (26):

$$C_{Q,t}={\displaystyle \frac{\Delta P_{loss,t}{\lambda }_{disc,t}}{\left|Q_t\right|}}$$

(26)

where $C_{Q,t}$ is the unit reactive-power compensation price in period t, computed by allocating the monetary value of the active-loss reduction $\Delta P_{loss,t}$ using the electricity price ${\lambda }_{disc,t}$ over the total reactive power $\left|Q_t\right|$. Then, the reactive power response subsidy $C_{Q,k}$ for an EV is expressed as Equation (27):

$$C_{Q,k}=\left|Q_{disc,t,k}\right|C_{Q,t}$$

(27)

where $Q_{disc,t,k}$ is the reactive power response of the k-th EV during period t.

4.2.2. Constraints

The charging power bounds, discharging power bounds, SOC bounds, and the SOC state-update equation for a single EV are given, respectively, by Equations (28)–(31):

$$P_{c,\min }<P_{c,t}<P_{c,\max }$$

(28)

$$P_{disc,\min }<P_{disc,t}<P_{disc,\max }$$

(29)

$$SO{C}_{\min }<SOC(t)<SO{C}_{\max }$$

(30)

$$SOC(t)=SOC(t-1)+{\displaystyle \frac{P_c(t)-P_{disc}(t)}{C}}$$

(31)

here, C is the battery capacity of the EV. The upper and lower limit constraints, set to maintain battery longevity by keeping the SOC within a specific range, confine the SOC between 0.2 and 0.8.

The charging-power ramp-rate and discharging-power ramp-rate constraints between consecutive periods are shown in Equations (32) and (33):

$$\left|P_{c,t+1}-P_{c,t}\right|\le \Delta P_{\max }$$

(32)

$$\left|P_{disc,t+1}-P_{disc,t}\right|\le \Delta P_{\max }$$

(33)

where $\Delta P_{\max }$ is the maximum allowable power ramp rate.

To ensure power quality and operational security of the grid, the node voltages and line power flows are constrained by Equations (34) and (35):

$$V_{\min }\le V_i^{(t)}\le V_{\max }$$

(34)

$$\left\{\begin{array}{c}{P}_{i,t}=U_{i,t}{\displaystyle \sum _{j=1}^N{U}_{j,t}(G_{ij}\cos {\theta }_{ij}+B_{ij}\sin {\theta }_{ij})}\\ Q_{i,t}=U_{i,t}{\displaystyle \sum _{j=1}^N{U}_{j,t}(G_{ij}\sin {\theta }_{ij}-B_{ij}\cos {\theta }_{ij})}\end{array}\right.$$

(35)

where $P_{i,t}$ and $Q_{i,t}$ are the active and reactive power loads at node i during period t, respectively; $U_{i,t}$ and $U_{j,t}$ are the voltages at node i and node j during period t, respectively; $G_{ij}$ and $B_{ij}$ are the conductance and susceptance between node i and node j, respectively; and ${\theta }_{ij}$ is the voltage phase angle difference between node i and node j during period t.

4.2.3. Multi-Objective Normalization

Conflicts and trade-offs exist among objective functions with different dimensions, making it difficult for all optimized metrics to simultaneously achieve their optimum values. Therefore, the range standardization method is employed to normalize the three optimization objectives into dimensionless forms. The expression is as Equation (36):

$$\begin{array}{cc}{y}_{ij}={\displaystyle \frac{x_{ij}-\underset{i}{\min }\left\{x_{ij}\right\}}{\underset{i}{\max }\left\{x_{ij}\right\}-\underset{i}{\min }\left\{x_{ij}\right\}}}& i=1,2,\cdots ,m;j=1,2,\cdots ,n\end{array}$$

(36)

where $y_{ij}$ is the dimensionless function data; $x_{ij}$ is the original value of the j-th objective function for the i-th candidate solution; $\underset{i}{\max }\left\{x_{ij}\right\}$ and $\underset{i}{\min }\left\{x_{ij}\right\}$ are the maximum and minimum values of the j-th objective function among all candidate solutions, respectively; m is the number of candidate solutions, and n is the number of objective functions. After normalizing the sub-objectives into dimensionless forms, the total objective function can be expressed as a normalized weighted sum, as shown in Equation (37):

$$\min F={\omega }_1{f}_1+{\omega }_2{f}_2+{\omega }_3{f}_3$$

(37)

4.3. Flowchart of the Proposed Two-Layer Optimization Model

To address the significant time-variability of RES output and EV loads in the distribution network, this section adopts a rolling optimization framework based on MPC to achieve joint optimization of multi-timescale EV charging/discharging schedules and dynamic network partitioning. The upper-layer model dynamically adjusts the distribution network partitioning on an hourly basis based on the WT/PV output and the dispatchable capacity of EVs for each period to achieve optimal resource dispatch in the spatial domain. The lower-layer model performs rolling optimization of EV charging/discharging schedules based on real-time states (with a 5 min time step) to achieve optimal power allocation in the temporal domain. The specific implementation steps of the proposed two-layer optimization model are outlined below, with its execution flowchart shown in Figure 6.

(1)

Initialization: Initialize the grid topology model, EV parameters, and forecasting WT/PV output curves. Set the scheduling resolution to $\Delta t=5$ min and initialize the time index $T=0$. Update the initial grid operating state and compute the initial partition used to start the closed-loop process.

(2)

Upper-layer hourly update (rectangle with an orange fill): The upper-level model is triggered at the beginning of each hour to dynamically adjust the network partitions. It (i) updates the electrical distance matrix using the forecasted wind/PV output and system states for the current period, (ii) performs dynamic partitioning as described in Chapter 3 to determine the hourly partition boundaries, and (iii) identifies the critical node within each partition by tracking the node with the largest voltage deviation. It then schedules the EVs that need to be connected to the grid in the next period to these critical nodes. The resulting partition scheme and critical nodes provide spatial guidance for the rolling control of the lower-level model in the following hour. It should be clarified that dynamic partitioning does not alter the physical network topology or reassign the connection points of already connected EVs. Once an EV is connected to the grid, it remains stationary. After the partition is dynamically adjusted in the next time period, newly arriving EVs in that subsequent period will be scheduled to connect to the updated critical node, while previously connected EVs remain unaffected.

(3)

Lower-layer rolling control and feedback (rectangle with a purple fill): At each 5 min interval, the controller checks the status of EVs. If new EVs are connected or existing ones are disconnected, a power flow update is performed and rolling-horizon optimization is triggered to generate updated charging/discharging and reactive power setpoints for all connected EVs over the forthcoming intervals. Otherwise, the setpoints from the previous rolling window continue to be used. After applying the current interval’s setpoints, the algorithm updates the equivalent load curve/state, sets $T=T+1$, and repeats until $T=T_{end}$. It is worth noting that the upper-layer partition update is executed hourly, while the lower-layer rolling optimization is triggered by EV connection/disconnection events, which reduces the number of full optimizations compared to strict periodic re-solving. Therefore, real-time deployability is evaluated at the level of a single rolling window: the time required to solve one rolling optimization should be shorter than the sampling interval. For larger systems, future adoption of computationally efficient power-flow approximations could further reduce the delay.

5. Simulation and Case Study Results

5.1. Simulation Parameter Settings

This section presents the simulation data and results to verify the applicability of the proposed method in distribution networks. The simulations are conducted in a MATLAB 2018b environment using a modified IEEE 33-node distribution system, whose topology is shown in Figure 7. The system includes four distributed PV generation systems and four wind turbines. The grid-connected charging and discharging efficiency of EVs and the power factor angle are set. The system base voltage, base capacity, and root node voltage are given. The EV battery capacity, charging/discharging power limit, SOC limits, and time resolution are set. The active power weight coefficient is also set. The specific simulation parameters are shown in

Table 1. Simulation Parameters.

Parameter Category	Parameters	Value
System Benchmark	Base Capacity	10 MVA
	Base Voltage	12.66 kV
	Reference Voltage of Root Node ($V_{ref}$)	1.0 p.u.
	Active Power Weight Coefficient ($\tau$)	0.7
	Time resolution	5 min
WT	Rated Capacity per WT	0.7 MW
	Total Installed Capacity	2.8 MW
	Access Nodes	[#17, #22, #24, #30]
	Power Factor	0.95
PV	Rated Capacity per PV Module	0.37 MW
	Total Installed Capacity	1.48 MW
	Access Nodes	[#5, #10, #14, #28]
	Power Factor	0.9
EV	Number of EVs Participating in Scheduling	100
	EV Charge Power Limit ($P_{c,\min }$; $P_{c,\max }$)	±7.5 kW
	EV Discharge Power Limit ($P_{disc,\min }$; $P_{disc,\max }$)	±7.5 kW
	EV Battery Capacity (C)	30 kWh
	Minimum SOC ($SO{C}_{\min }$)	0.2
	Maximum SOC ($SO{C}_{\max }$)	0.8
	EV charging/discharging efficiency	0.9
	max power factor angle (${\mathrm{\phi }}_{\min }$)	−arccos(0.95)
	min power factor angle (${\mathrm{\phi }}_{\max }$)	+arccos(0.95)

.

To clarify the validation scope, the modified IEEE 33-node feeder is adopted as a standard benchmark for distribution-network voltage studies, enabling controlled comparison and reproducibility. The EV fleet size is selected to create a meaningful voltage-regulation stress level rather than to mirror a full urban population: with 100 EVs participating and a per-EV active power limit of ±7.5 kW, the aggregator-level controllable active power capacity reaches up to 0.75 MW, and the available energy capacity is 3 MWh (30 kWh × 100). Relative to the installed RES capacity in this test system (WT 2.8 MW and PV 1.48 MW), this penetration is sufficient to produce observable voltage excursions under uncoordinated charging and to evaluate the benefits of coordinated active–reactive scheduling. The model in this paper employs the Gurobi solver, with the maximum computational time for a single MPC optimization being less than 10 s, which is significantly acceptable for the 5 min-level rolling optimization.

The typical daily WT and PV output curves are shown in Figure 8:

5.2. Dynamic Network Partitioning Results

According to the method described in Section 3, the electrical distance matrix is reconstructed hourly based on the current WT and PV output, and optimal partitioning is carried out considering the time-varying EV capacity. This generates 24 sets of partitioning results for an entire day. The partitioning results for typical time periods are shown in the figure below. Here, node #1 is the slack (balance) node. It does not participate in the partitioning process and is directly assigned to the partition containing the node to which it is most closely connected.

Figure 9 shows the partitioning results at four typical time slots: 1:00 a.m., 2:00 a.m., 4:00 p.m., and 6:00 p.m. It can be observed that the nodes within each partition across different time periods are geographically closely connected, with no isolated nodes or unreasonable long-distance connections. Furthermore, the partition boundaries adjust according to changes in electrical distance, which support the validity of the rationality of the comprehensive electrical distance matrix used.

Furthermore, as described by the model introduced in this paper, after partitioning at each time step, the node with the maximum voltage deviation within a region is designated as the charging/discharging node. EVs in that region that connect to the grid during the corresponding period are scheduled to reach this node, as indicated by the green nodes in Figure 9. It can be seen that the location of the node with the maximum voltage deviation within a region dynamically changes over time after partitioning, indicating that the partitioning strategy can respond to fluctuations in system operating states. By dynamically adjusting the connection points of EVs, the flexible charging and discharging capabilities of EVs can be maximized. Moreover, leveraging the spatio-temporal transfer characteristics of EVs mitigates voltage violations and enhances the accommodation capability of renewable energy.

5.3. Distribution Network Voltage Optimization Analysis

5.3.1. Impact of Controlled Versus Uncontrolled Electric Vehicle Charging on Grid Voltage

The number of EVs integrated into the distribution network is set to 100. All connected EVs are assumed to be enrolled in the coordinator and to follow the computed charging/discharging and reactive power setpoints during their plug-in windows [46]. This assumption is used to quantify the upper-bound technical potential of the proposed optimization framework under a fixed controllable capacity, as modeling opt-out behavior driven by user preferences is outside the scope of this study. EV mobility requirements are still protected by the SOC dynamics and departure SOC constraints in Equations (28)–(31), so discharging is only scheduled when the required SOC at departure is satisfied. Figure 10 displays the voltage profiles of nodes #18 and #33 under three conditions: (i) no EV integration, (ii) uncoordinated EV charging (operating at the maximum power-factor angle limit), and (iii) EV-responsive operation.

Figure 10 presents a comparison of node voltage profiles under three distinct operational scenarios: the blue curve corresponds to the baseline case with no EV integration; the red curve represents the scenario of uncoordinated EV charging (i.e., charging at the maximum power-factor angle limit); and the yellow curve depicts the coordinated EV charging/discharging response mode proposed in this paper. It can be observed that the integration of EV charging loads into the distribution network causes a noticeable voltage drop at the end-of-line nodes. The magnitude of this voltage drop is further exacerbated during the EV charging peak period from 20:00 to 24:00, which may pose a significant threat to the steady-state operational security of the power grid. Specifically, compared to the baseline scenario without EV integration, uncoordinated charging introduces additional load peaks, leading to more severe voltage drops and heightening the risk of node voltage falling below the lower limit. In contrast, the coordinated charging/discharging strategy proposed in this paper, by regulating both EV active power and the power factor angle, can effectively elevate the voltage level at the end nodes. The optimized voltage profile is overall smoother with significantly reduced fluctuations, thereby enhancing voltage quality while substantially lowering the probability of node voltage violations.

To provide a more comprehensive evaluation of the proposed strategy’s effectiveness, Figure 11 and Figure 12 further illustrate the system-wide voltage distribution across the entire IEEE 33-node distribution network before and after optimization.

Figure 11 and Figure 12 present the three-dimensional spatiotemporal voltage distribution surfaces for the IEEE 33-bus system over a full day. The horizontal axes represent time and bus number, while the vertical axis denotes the voltage magnitude V (p.u.). The color bar provides a voltage map, with blue indicating higher voltages and red indicating lower voltages. Each fixed time point corresponds to the voltage distribution across all buses along the entire feeder at that moment, whereas each fixed bus shows the voltage variation trajectory of an individual bus throughout the day. Under the uncoordinated charging mode (shown in Figure 11), extensive low-voltage depressions occur across many buses and time periods, with the most severe voltage drops concentrated at electrically weak/downstream buses. This indicates that random, concentrated EV charging exacerbates voltage sag and increases the risk of violating the lower voltage limit. In contrast, under the coordinated charging optimization mode (shown in Figure 12), the low-voltage area is significantly reduced, and the entire surface is shifted upward. This demonstrates that by performing spatiotemporal scheduling and coordinated power optimization of EV charging behavior, the coordinated charging strategy can effectively improve voltage uniformity among buses, mitigate voltage depression, and enhance the overall system’s voltage security, thereby validating the effectiveness of the proposed strategy in enhancing system voltage stability.

5.3.2. Voltage Comparison Between Dynamic and Static Charging Nodes

To further investigate the impact of varying EV connection points on distribution network voltage, this section conducts a comparative experiment using the following two scheduling strategies: (a) dynamic partitioning, which selects charging/discharging nodes within each partition, and (b) fixed-node scheduling, which schedules only the EVs connected to predetermined charging nodes.

Strategy #1: Incorporates the dynamic partitioning strategy proposed in this paper, where EV connection points change according to the partition adjustments.

Strategy #2: Adopts the traditional fixed connection node strategy, selecting nodes #13, #23, and #29 as the EV connection points.

The voltage profiles of key nodes under the two strategies are shown in Figure 13 and Figure 14. Based on the voltage trajectories, when strategy #1 is employed, the degree of voltage rise at the end nodes is significantly greater than that under strategy #2. This indicates the superiority of dynamically selecting connection points in distribution networks with limited dispatchable EV resources for voltage support.

5.3.3. Voltage Comparison with and Without Renewable Energy Forecast Errors

Baseline simulations assume accurate short-term WT and PV forecasts in order to focus on the control and optimization performance of the proposed framework. To evaluate robustness against renewable forecast uncertainty, an additional sensitivity test was conducted by perturbing both WT and PV forecasts with zero-mean Gaussian noise [46]. The standard deviation of the forecast error was set to 30% of the corresponding renewable power, representing a high-uncertainty scenario commonly reported in short-term RES forecasting. Figure 15 and Figure 16 compare the PV and WT output curves with and without high forecasting errors.

As shown in Figure 17 and Figure 18, under the scenario with relatively high forecasting errors of WT and PV output, the voltage at the end-node exhibits a slight overall decrease compared to the case with accurate renewable generation. However, the two coordinated optimization voltage curves corresponding to the two forecasting scenarios, respectively, remain nearly identical throughout the day, with only minimal discrepancy. In contrast, the uncoordinated charging curve still leads to a more pronounced and sustained voltage drop during the evening charging peak. This indicates that, under the given renewable penetration level and high forecasting errors in this case study, the proposed coordinated scheduling strategy demonstrates good robustness against WT/PV output errors, and the voltage regulation performance is still predominantly driven by the coordinated active and reactive power adjustment capability of EVs. This robustness stems primarily from the hierarchical control structure of the proposed framework: the lower-layer 5 min rolling optimization incorporates feedback correction, while the upper-layer partitioning is updated hourly rather than relying on instantaneously accurate values, which effectively suppresses the amplification of moderate noise in the voltage profile. Meanwhile, the combined active–reactive power regulation from EVs provides the main support for voltage, thereby absorbing WT/PV output fluctuations within a certain range. Consequently, despite significant forecasting uncertainty, the proposed method maintains voltage regulation performance close to that of the ideal scenario.

5.4. Network Loss Optimization Analysis

This section analyzes the optimization effect on network losses by comparing the trends of total network losses under different operating scenarios. Figure 19 shows the comparison of the total network loss over a 24 h period before and after the process optimization, while Figure 20 shows network loss savings achieved by the coordinated strategy compared to the uncoordinated charging mode.

In the illustrated results, the green curve represents the total network losses after the proposed optimization, the red curve corresponds to the total network losses under the uncoordinated charging scenario, and the blue curve indicates the baseline network losses with no EV integration. It can be observed that the coordinated optimization model proposed in this paper effectively suppresses the increase in network losses. At the system level, this strategy achieves a total network loss reduction of 4991.3 kW, representing a 3.9% decrease compared to the uncoordinated charging mode. To further investigate the spatio-temporal evolution of network losses, Figure 21 and Figure 22 present the time-varying loss profiles of end-of-line nodes in the distribution network over a 24 h period.

From the network loss curves of the end-of-line nodes, it can be observed that the optimized loss level is significantly lower than that of the uncoordinated charging scenario and generally fluctuates below the baseline loss line in the no-EV-integration case. This validates that the proposed strategy has a significant inhibitory effect on the additional network losses caused by uncoordinated EV integration. It also indicates that leveraging the reactive power response capability of EVs can reduce system network losses to a certain extent. It is noteworthy that the optimized loss curve exhibits fluctuation, primarily attributable to the dynamic nature of the scheduling strategy. The strategy dynamically schedules EVs within a partition to the node with the maximum voltage deviation in that region on an hourly basis, thereby relaxing the spatial fixed EV connection points. Natural fluctuations in system load, the intermittency of renewable energy output, and real-time changes in the status of the EV fleet continuously alter power flows and node voltages. Due to differences in electrical parameters and local network structure among nodes, each targeted dispatch triggers a redistribution of on-site power flow. Furthermore, to rapidly support voltage-weak nodes, dispatch commands often result in concentrated power injection or absorption at the target nodes. This abrupt change in short-term power further contributes to instantaneous fluctuations in network losses. It should be pointed out that such fluctuations are an inherent characteristic of the dynamic scheduling strategy, which targets precise and rapid voltage support and loss suppression. They reflect responsiveness to the real-time system state and do not detract from its overall optimization effectiveness.

To further investigate the impact of EV connection points on distribution network losses, the network loss curves under the two strategies defined in Section 5.3.2 are compared, as shown in Figure 23.

In the illustrated results, the green curve corresponds to strategy #1 (dynamic location optimization) described in Section 5.3.2, while the purple curve corresponds to strategy #2 (fixed location optimization). It can be observed that the total system losses under strategy #1 are further reduced by 0.7% compared with strategy #2. This demonstrates that, in addition to coordinated charging and discharging, dynamically adjusting the connection points provides additional loss-reduction benefits. In terms of temporal characteristics, the loss curves under both strategies exhibit similar variation trends aligned with the time-varying pattern of system load. However, the loss curve under dynamic location scheduling shows more pronounced fluctuations. The root cause of this difference lies in the fact that the fixed-location strategy executes power regulation only at predetermined nodes, resulting in a relatively static zone of influence. In contrast, the dynamic-location strategy dynamically selects and switches charging/discharging node positions according to the system’s real-time voltage profile and load state. This location adaptability enables power regulation to respond more precisely to time-varying voltage-weak nodes. Although the increased control freedom may introduce larger short-term fluctuations, it also more fully unleashes the potential for loss reduction. This result validates that incorporating dynamic node selection into charging/discharging scheduling can effectively enhance the economic performance of system operation. It also suggests that dispatching EVs to nodes with high network losses or weak voltage can reduce the losses associated with long-distance power transmission. Simultaneously, the reactive power support provided by EVs also contributes to reducing the additional losses caused by reactive power flow along distribution lines.

6. Summary and Prospect

6.1. Summary

This paper proposes a distribution network voltage stability optimization framework that integrates dynamic partitioning and coordinated electric vehicle scheduling. By updating the electrical distance matrix in real-time, it achieves hourly dynamic partitioning, effectively adapting to renewable energy output fluctuations and the spatio-temporal transfer characteristics of EVs. By controlling the power factor angle of the charger, it enables coordinated optimization of active EV charging/discharging and reactive power response, fully exploiting EV’s regulatory potential as a flexible bidirectional resource. A hierarchical optimization framework based on MPC achieves synergy between spatial resource localization and temporal power allocation. Simulation results demonstrate that the proposed method significantly improves the uniformity of the distribution network voltage profile, eliminates the risk of voltage violations, and effectively reduces system network losses. It provides an effective solution for the secure and economical operation of distribution networks under high-penetration renewable energy integration.

6.2. Discussion in Real-World Application

The proposed dynamic partitioning and coordinated scheduling framework has been primarily validated on a modified IEEE 33-node system at the feeder level. Due to its decomposition of regulation tasks in both spatial and temporal dimensions, the framework is structurally compatible with large-scale systems. Specifically, when employing commercial solvers such as Gurobi, the optimization time for a single time step is on the order of seconds, which is fully acceptable relative to the 5 min real-time scheduling cycle. For large-scale systems involving thousands of nodes, the dynamic partitioning mechanism allows the network to be divided into a larger number of finer, locally autonomous zones. The proposed coordinated scheduling method can then be independently deployed and operated within each zone, enabling scalable management of complex systems.

Effective real-world implementation requires the following: (1) The distribution network must have a well-established measurement and communication infrastructure capable of supporting the real-time perception of states such as node voltages and branch power flows, as well as the rapid dissemination of partition instructions. (2) There needs to be a certain scale and spatiotemporally concentrated EV fleet within the network area to ensure that each partition possesses sufficient dispatchable resources for localized voltage support. (3) Short-term forecasts of wind and PV output, as well as EV behavior models, must be constructed based on long-term, reliable historical data, as the adaptability of the partitioning and scheduling schemes is directly influenced by forecast accuracy. (4) Effective market or incentive mechanisms must be established to ensure that EV users are willing to authorize coordinated regulation of their vehicles’ charging/discharging power and power factor angle, provided their travel needs are met. While the proposed optimization framework assumes that appropriate incentives exist to secure user participation, the explicit design of dynamic pricing or market mechanisms that jointly reward both active and reactive power services is not addressed in this study and remains an important direction for future research.

Furthermore, in practical deployment, issues such as communication delays, data synchronization errors, and the consistency of user response to control commands will directly impact the effectiveness of the closed-loop control. These engineering implementation challenges can serve as directions for future research. In addition, although reactive power regulation does not induce battery cycling, frequent adjustment of converter operating points may contribute to thermal stress in power electronic components, which is not the main focus of this study. However, the proposed control operates on a minute-level timescale with bounded power factor angles, and practical deployments can further limit stress through ramp-rate constraints or deadband designs. Detailed lifespan and aging effect modeling of EV power electronics is left for future work.

6.3. Future Work

However, as renewable energy penetration continues to increase and research on vehicle–grid interaction deepens, future work based on the dynamic partitioning and coordinated scheduling framework established in this paper should focus on strengthening scalability validation and computational practicality. The proposed hierarchical framework should be evaluated on larger and more realistic distribution feeders under higher EV penetration levels to quantify how solution time and control performance scale with network size and EV population. In the temporal dimension, faster multi-resource coordinated control at finer temporal resolutions can be explored to respond to rapid fluctuations in renewable energy sources and loads. In the resource dimension, considering the complexity of actual distribution network resources, diverse distributed resources such as energy storage systems and flexible loads can be integrated into a unified collaborative optimization framework to build a more comprehensive architecture. Finally, to address the computational and communication burdens in large-scale deployments, scalable implementation approaches based on distributed optimization, warm-starting, and computationally efficient power flow models should be investigated to enhance engineering robustness.